In the application of physics-informed neural networks (PINNs) for solutions of partial differential equations, the optimizer may fall into a bad local optimal solution during the training of the network. In this case, the shape of the desired solution may deviate from that of the real solution. To address this problem, we have combined the priori information and knowledge transfer with PINNs. The physics-informed neural networks with a priori information (pr-PINNs) were introduced here, which allow the optimizer to converge to a better solution, improve the training accuracy, and reduce the training time. For the experimental examples, different kinds of localized wave solutions for several types of Korteweg-de Vries (KdV) equations were solved using pr-PINNs. Multi-soliton solutions of the KdV equation, multi-soliton and lump solutions of the (2+1)-dimensional KdV equation, and higher-order rational solutions of the combined KdV-mKdV equation have been solved by pr-PINNs. By comparing the results of pr-PINNs with PINNs under the same configuration, pr-PINNs show higher accuracy and lower cost in solving different solutions of nonlinear evolution equations due to the combination of the priori information with PINNs, which enables the neural network to capture the characteristics of the solution during training. The good performance of the proposed method will have important potential application value for the solutions of real-world problems.
Citation: Zhi-Ying Feng, Xiang-Hua Meng, Xiao-Ge Xu. The data-driven localized wave solutions of KdV-type equations via physics-informed neural networks with a priori information[J]. AIMS Mathematics, 2024, 9(11): 33263-33285. doi: 10.3934/math.20241587
In the application of physics-informed neural networks (PINNs) for solutions of partial differential equations, the optimizer may fall into a bad local optimal solution during the training of the network. In this case, the shape of the desired solution may deviate from that of the real solution. To address this problem, we have combined the priori information and knowledge transfer with PINNs. The physics-informed neural networks with a priori information (pr-PINNs) were introduced here, which allow the optimizer to converge to a better solution, improve the training accuracy, and reduce the training time. For the experimental examples, different kinds of localized wave solutions for several types of Korteweg-de Vries (KdV) equations were solved using pr-PINNs. Multi-soliton solutions of the KdV equation, multi-soliton and lump solutions of the (2+1)-dimensional KdV equation, and higher-order rational solutions of the combined KdV-mKdV equation have been solved by pr-PINNs. By comparing the results of pr-PINNs with PINNs under the same configuration, pr-PINNs show higher accuracy and lower cost in solving different solutions of nonlinear evolution equations due to the combination of the priori information with PINNs, which enables the neural network to capture the characteristics of the solution during training. The good performance of the proposed method will have important potential application value for the solutions of real-world problems.
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Zhi-Ying Feng, Xiang-Hua Meng, Xiao-Ge Xu. The data-driven localized wave solutions of KdV-type equations via physics-informed neural networks with a priori information[J]. AIMS Mathematics, 2024, 9(11): 33263-33285. doi: 10.3934/math.20241587
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